Some Continued Fractions in Ramanujan’s Lost Notebook

The connections between the Golden Ratio namely (1 + √5)/2, a simple continued fraction, and Fibonacci and Lucas numbers, are familiar. The Fibonacci and Lucas numbers have many fascinating properties. We now point out that the square root of the Golden Ratio is the real part of a simple periodic continued fraction but using (complex) Gaussian integers a + ib instead of the natural integers. This fact provokes a definition and a study of complex Fibonacci and Lucas numbers, and the study again turns out to have a rich theoretic structure. A fuller account will appear in The Fibonacci Quarterly.

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

This is a bibliographical review that provides a brief glimpse into the beauty of continued fractions and how they can be very useful, both in obtaining good rational approximations of a given real number and in their representation as a continued fraction. Furthermore, it presents important properties, such as the relationship between quadratic irrationals and periodic continued fractions. On the other hand, with a view to a potential introduction of this topic in elementary education, a method for obtaining square root approximations through continued fractions is presented. Finally, using more advanced tools, we present an infinite continued fraction representation of the Euler number, which consequently implies the irrationality of e.

Периодичность и квазипериодичность функциональных непрерывных дробей в гиперэллиптическом поле $$L = \mathbb{Q}(x)(\sqrt{f})$$ имеет более сложную природу, чем периодичность числовых непрерывных дробей элементов квадратичных полей. Известно, что периодичность непрерывной дроби элемента $$\sqrt{f}/h^{g+1}$$, построенной по нормированию, связанному с многочленом h первой степени, эквивалентна наличию нетривиальных S-единиц в поле L рода g и эквивалентна наличию нетривиального кручения в группе классов дивизоров. В данной статье найден точный промежуток значений $$s \in \mathbb{Z}$$ таких, что элементы $$\sqrt{f}/h^s$$ имеют периодическое разложение в непрерывную дробь, где $$f \in \mathbb{Q}[x]$$ - свободный от квадратов многочлен четной степени. Для многочленов f нечетной степени проблема периодичности непрерывных дробей элементов вида $$\sqrt{f}/h^s$$ рассмотрена в статье [5], причем там доказано, что длина квазипериода не превосходит степени фундаментальной S-единицы поля L. Проблема периодичности непрерывных дробей элементов вида $$\sqrt{f}/h^s$$ для многочленов f четной степени является более сложной. Это подчеркивается найденным нами примером многочлена f степени 4, для которого соответствующие непрерывные дроби имеют аномально большую длину периода. Ранее в статье [5] также были найдены примеры непрерывных дробей элементов гиперэллиптического поля L с длиной квазипериода значительно превосходившей степень фундаментальной S-единицы поля L.

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction F ( q ) F(q) . In order to prove some of these claims, G. N. Watson established two important theorems about F ( q ) F(q) that he found in Ramanujan’s notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

Ramanujan, who loved continued fractions, recorded many of his formulas in his two notebooks. The 25th entry in Chapter 12 of Notebook II is a continued fraction involving a quotient of gamma function products. We are going to give a new proof of Entry 25, which, incidently, is the least difficult of all Ramanujan's continued fraction formulas involving quotients of gamma function products. In our proof, we make use of a hypergeometric formula of his.

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

When a particle dimension is comparable to the laser light wavelength, scattering calculations require the use of Mie or similar infinite series. The terms in the series may be an infinite set of complex determinants. Despite the power of modern computers these equations are difficult to compute with, because of the lack of analytical criteria for terminating the computations. Commonly, the series computations are stopped when the value of the series is changed by an amount less than a predetermined factor as a result of the addition of the computed term. This procedure has led to large errors. These series are easily transformed into the continued fraction form of the Fsad6 approximates. Complex continued fractions are more stable, computationally, than infinite series and frequently converge more rapidly. Further, an anlytical test exists for determining the circular domain within which the value of the continued fraction must lie. This permits an accurate and precise delineation of the maximum residual error. There is a disadvantage in the use of continued fractions. If the order of the creates convergent is not great enough for the desired accuracy, the computation must be repeated. This is not a serious handicap in actual use.

Ramanujan recorded many beautiful continued fractions in his notebooks. In this paper, we derive several indentities involving the Ramanujan continued fraction c(q), including relations between c(q) and c(qn). We also obtain explicit evaluations of for various positive integers n.

defined over the field of rational numbers Q is normal. They do this by noting that, over a field of characteristic 3, F (X) equals 1/ √ 1 + 1/X. The latter has a periodic continued fraction and, as a consequence, they are able to explicitly verify that F (X) is normal. They then use an “automatic” proof to show that the Laurent series of F (X), defined over Q, is normal. Here we shall give a different proof of the latter result (Theorem 4, below) and show, more generally, that if a formal Laurent series Γ (X) defined over Q is defined and normal (mod p) for any prime p, then it is also normal over Q. We shall also consider the continued fraction of the more general surd 1/ √ 1 + u/X + v/X2 over any field not of characteristic 2. Put δ = u/4−v and note that √ 1 + u/X + v/X2 = 1 + u/(2X) when δ = 0. Hence, when δ = 0, the continued fraction expansion is trivial and is not normal. Define

In the classical case, the connection between the periodicity of the continued fraction of $$\sqrt f $$ and the existence of a fundamental unit of the corresponding hyperelliptic field $$\mathcal{L} = K(x)(\sqrt f )$$ , where K is a field of characteristic different from 2, has long been known. For the element $$\sqrt f $$ , the period length of the continued fraction in $$K((1{\text{/}}x))$$ can be trivially estimated from above by the doubled degree of the fundamental unit. Much more complicated and interesting is the problem of estimating (from above) the period length of other elements of $$\mathcal{L}$$ that have a periodic continued fraction. Among these elements, those of the form $$\sqrt f {\text{/}}{{x}^{s}}$$ , $$s \in \mathbb{Z}$$ , play a key role. For such elements, the period length can be many times greater than the double degree of the fundamental unit. In this article, we find upper bounds for the period length of key elements of hyperelliptic fields $$\mathcal{L}$$ over number fields K. An example is found that demonstrates the sharpness of the proven upper bounds.

Domains of validity for some of Ramanujan's continued fraction formulas

Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β 1 y 1 + β 2 y 2 + α| is given, where y 1 = y(x 1), y 2 = y(x 2), x 1 and x 2 are positive integers, α, β 1 and β 2 are algebraic irrational numbers.

O přibližných hodnotách řetězce se stálým jmenovatelem

Let K ( a n / b n ) K({a_n}/{b_n}) be a limit periodic continued fraction of elliptic type; i.e., a n → a {a_n} \to a and b n → b {b_n} \to b , where a / ( b + w ) a/(b + w) is an elliptic linear fractional transformation of w . We show that if ∑ | a n − a | > ∞ \sum {|{a_n} - a| > \infty } and ∑ | b n − b | > ∞ \sum {|{b_n} - b| > \infty } , then K ( a n / b n ) K({a_n}/{b_n}) diverges. This generalizes the well-known Stern-Stolz Theorem. The Gauss continued fraction (related to hypergeometric functions) is used as an example. We also give an example where a n − a = O ( n − 1 ) {a_n} - a = \mathcal {O}({n^{ - 1}}) and b n = b = 1 {b_n} = b = 1 and K ( a n / b n ) K({a_n}/{b_n}) converges. The divergence result is also generalized further.